فهرست مطالب
Computational Methods for Differential Equations
Volume:6 Issue: 2, Spring 2018
- تاریخ انتشار: 1397/01/12
- تعداد عناوین: 9
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Pages 111-127In this paper, we present new variants of global bi-conjugate gradient (Gl-BiCG) and global bi-conjugate residual (Gl-BiCR) methods for solving nonsymmetric linear systems with multiple right-hand sides. These methods are based on global oblique projections of the initial residual onto a matrix Krylov subspace. It is shown that these new algorithms converge faster and more smoothly than the Gl-BiCG and Gl-BiCR methods. The preconditioned versions of these methods are also explored in this study. Eventually, the efficiency of these approaches are demonstrated through numerical experimental results arising from two and three-dimensional advection dominated elliptic PDE.Keywords: Matrix Krylov subspaces, Elliptic Partial differential equation, Non symmetric linear systems, Global iterative methods, Multiple right-hand sides
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Pages 128-140In this article, we propose the definition of one parameter matrix Mittag-Leffler functions of fractional nabla calculus and present three different algorithms to construct them. Examples are provided to illustrate the applicability of suggested algorithms.Keywords: Fractional order, Nabla difference, Mittag-Leffler function, Spectral radius, $N$-transform
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Pages 141-156The method of quasilinearization is an effective tool to solve nonlinear equations when some conditions on the nonlinear term of the problem are satisfied. When the conditions hold, applying this technique gives two sequences of coupled linear equations and the solutions of these linear equations are quadratically convergent to the solution of the nonlinear problem. In this article, using some transformations, the well-known Blasius equation which is a nonlinear third order boundary value problem, is converted to a nonlinear Volterra integral equation satisfying the conditions of the quasilinearization scheme. By applying the quasilinearization, the solutions of the obtained linear integral equations are approximated by the collocation method. Employing the inverse of the transformation gives the approximation solution of the Blasius equation. Error analysis is performed and comparison of results with the other methods shows the priority of the proposed method.Keywords: Quasilinearization technique, Volterra integral equations, Blasius equation, Collocation method
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Pages 157-175In this paper we establish a new numerical method for solving a class of stochastic partial differential equations (SPDEs) based on B-splines wavelets. The method combines implicit collocation with the multi-scale method. Using the multi-scale method, SPDEs can be solved on a given subdomain with more accuracy and lower computational cost than the rest of the domain. The stability and consistency of the method are provided. Also numerical experiments illustrate the behavior of the proposed method.Keywords: Multi-scale method, Cubic B-splines, Stochastic partial differential equations
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Pages 176-185In this article, we will show the complex inversion formula for the inversion of the L2-transform and also some applications of the L2, and Post Widder transforms for solving singular integral equation with trigonometric kernel. Finally, we obtained analytic solution for a partial differential equation with non-constant coefficients.Keywords: L2-Transform, Laplace transform, Complex inversion formula, Post-Widder transform
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Pages 186-214In this paper, we propose the spectral collocation method based on radial basis functions to solve the fractional Bagley-Torvik equation under uncertainty, in the fuzzy Caputo's H-differentiability sense with order ($1< nu < 2$). We define the fuzzy Caputo's H-differentiability sense with order $nu$ ($1< nu < 2$), and employ the collocation RBF method for upper and lower approximate solutions. The main advantage of this approach is that the fuzzy fractional Bagley-Torvik equation is reduced to the problem of solving two systems of linear equations. Determining a good shape parameter is still an outstanding research topic. To eliminate the effects of the radial basis function shape parameter, we use thin plate spline radial basis functions which have no shape parameter. The numerical investigation is presented in this paper shows that excellent accuracy can be obtained even when few nodes are used in analysis. Efficiency and effectiveness of the proposed procedure is examined by solving two benchmark problems.Keywords: The fractional Bagley Torvik equation, Meshless method, RBF collocation, Thin plate splines, Fuzzy Caputo', s H-differentiability
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Pages 215-237A meshless numerical technique is proposed for solving the generalized variable coefficient Schrodinger equation and Schrodinger-Boussinesq system with electromagnetic fields. The employed meshless technique is based on a generalized smoothed particle hydrodynamics (SPH) approach. The spatial direction has been discretized with the generalized SPH technique. Thus, we obtain a system of ordinary differential equations (ODEs). Also, it is clear in the numerical methods for solving the time-dependent initial boundary value problems, based on the meshless methods, to achieve the high-order accuracy the temporal direction must be solved using an effective technique. Thus, in the current paper, we apply the fourth-order exponential time differenceing Runge-Kutta method (ETDRK4) for the obtained system of ODEs. The aim of this paper is to show that the meshless method based on the generalized SPH approach is suitable for the treatment of the nonlinear complex partial differential equations. Numerical examples confirm the efficiency of proposed scheme.Keywords: Schrodinger-Boussinesq system, Meshless method, Smoothed particle hydrodynamic method, Fourth-order exponential time differenceing Runge-Kutta method
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Pages 238-247In this paper, we improve b-spline collocation method for Benjamin-Bona-Mahony-Burgers (BBMB) by using defect correction principle. The exact finite difference scheme is used to find defect and the defect correction principle is used to improve collocation method. The method is tested on somemodel problems and the numerical results have been obtained and compared.Keywords: Exact finite difference, Defect principle, Collocation method, Deviation of the error
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Pages 248-265This paper deals with a ratio-dependent functional response predator-prey model with a fractional order derivative. The ratio-dependent models are very interesting, since they expose neither the paradox of enrichment nor the biological control paradox. We study the local stability of equilibria of the original system and its discretized counterpart. We show that the discretized system, which is not more of fractional order, exhibits much richer dynamical behavior than its corresponding fractional order model. Specially, in the discretized system, many types of bifurcations (transcritical, flip, Neimark-Sacker) and chaos may happen, however, the local analysis of the fractional-order counterpart, only deals with the stability (unstability) of the equilibria. Finally, some numerical simulations are performed by MATLAB, to support our analytic results.Keywords: Ratio-dependent functional response model, Fractional derivative, Discretization, Bifurcation, chaos